What exactly is meant by this? That she is visiting me, or she rents a room from me, or can be both depending on the context? In those cases, particularly the first, I usually would say "She is staying with us". Can ở trọ mean that?
When you say "Cô ấy ở trọ với tôi.", that means she and you rent a same room from someone, and then you both pay for that room together.
The verb "lodge" isn't really used in speech in English. I've heard of "lodge" and "lodging"as nouns, but we don't say we're lodging somewhere.
In this case, you would say: "Cô ấy ở trọ nhà tôi." Or "Cô ấy ở trọ tại nhà tôi." (It means she rents a room in my house.)